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If $k$ is any natural number, then does there exist a sequence $(p_n)$ of primes such that for each $n \in \mathbb{N}$, $k|(p_n +1)$?

It would be equally good if one can prove/disprove the above question with the condition that $k|(p_n-1)$ instead of $k|(p_n+1)$.

I don't even know how to start thinking about this problem and so I decided post it here. Thanks in advance.

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up vote 2 down vote accepted

Your condition I is just to say that $p_n\equiv -1\pmod k$, and condition II: $p_n\equiv 1\pmod k$. By Dirichlet theorem, this is true in any case.
P.S. I think this is just what Andres Caicedo meant, when he gave the link.

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